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# Level Lowering Modulo Prime Powers and Twisted Fermat Equations

Published:2011-09-15
Printed: Apr 2012
• Sander R. Dahmen,
Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2
• Soroosh Yazdani,
Department of Mathematics and Statistics, McMaster University, West Hamilton, ON, L8S 4K1
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## Abstract

We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$.
 Keywords: modular forms, level lowering, Diophantine equations
 MSC Classifications: 11D41 - Higher degree equations; Fermat's equation 11F33 - Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11F11 - Holomorphic modular forms of integral weight 11F80 - Galois representations 11G05 - Elliptic curves over global fields [See also 14H52]

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